Convection and Conduction Heat Transfer Part 8 pot

2.1 Effective thermal conductivity Composite materials typically consist of stiff and strong material phase, often as fibres, held together by a binder of matrix material, often an organ

2 Fibrous composite material

In the present paper, a composite material consisting of two materials is analysed It is a

fibrous material with unidirectional fibres The material of the matrix is homogenous and its

thermal conductivity is constant Fibres are also homogenous, however, they may differ

from each other when it comes to radius or thermal conductivity

2.1 Effective thermal conductivity

Composite materials typically consist of stiff and strong material phase, often as fibres, held

together by a binder of matrix material, often an organic polymer Matrix is soft and weak,

and its direct load bearing is negligible In order to achieve particular properties in preferred

directions, continuous fibres are usually employed in structures having essentially two

dimensional characteristics

Applying the fundamental definition of thermal conductivity to a unit cell of unidirectional

fibre reinforced composite with air voids, one can deduce simple empirical formula to

predict the thermal conductivity of the composite material with estimated air void volume

percent (Al-Sulaiman et al., 2006) The ability to accurately predict the thermal conductivity

of composite has several practical applications The most basic thermal-conductivity models

(McCullough, 1985) start with the standard mixture rule

and inverse mixture rule

fraction of i-th composite constituents (e.g resin, fibre, void)

The composite thermal conductivity in the filler direction is estimated by the rule of

mixtures The rule of mixtures is the weighted average of filler and matrix thermal

conductivities This model is typically used to predict the thermal conductivity of a

unidirectional composite with continuous fibres In the direction perpendicular to the fillers

(through plane direction), the series model (inverse mixing rule) is used to estimate

composite thermal conductivity of a unidirectional continuous fibre composite

Another model similar to the two standard-mixing rule models is the geometric model (Ott,

1981)

Numerous existing relationships are obtained as special cases of above equations Filler

shapes ranging from platelet, particulate, and short-fibre, to continuous fibre are

consolidated within the relationship given by McCullough (McCullough, 1985)

The effective thermal conductivity for a composite solid depends, however, on the geometry

assumed for the problem In general, to calculate the effective thermal conductivity of

fibrous materials, we have to solve the energy transport equations for the temperature and

heat flux fields For a steady pure thermal conduction with no phase change, no convection

and no contact thermal resistance, the equations to be solved are a series of Poisson

equations subject to temperature and heat flux continuity constraints at the phase interfaces

After the temperature field is solved, the effective thermal conductivity, λeff, can be

determined

where q is the steady heat flux through the cross-section area dA between the temperature

difference ΔT on a distance L Heat flow through the unit area of the surface with normal n

is linked with the temperature gradient in the n-direction by Fourier's law as

(5)

2.2 Composite structure

The elementary cell of the considered composite is a cross-sectional square and it is

perpendicular to fibres direction Perfect contact between the matrix and the cell is assumed,

heat transfer does not depend on time, and only conductive transfer is considered Also,

none of materials’ properties depends on temperature, so the problem is linear and can be

described by Laplace equation in each domain

Fig 1 Composite elementary cell structure

Governing equation of the problem both in the matrix domain and in each fibre domain

takes the following form:

Boundary condition applied to the cell are defined as follows:

TM

TC 290K y 0, (8)

refer to the matrix and fibres

Hence, one can see that the composite is heated from the bottom and cooled from the above

Symmetry condition is applied on the sides of the cell, which means that the heat flux on

these boundaries equals zero Thermal continuity and heat flux continuity conditions are

applied on the boundary of each fibre

2.3 Relation between geometry and conductivity

As we have already mentioned, the geometrical structure of the composite material may

have a great impact on the resultant effective conductivity of the composite Commonly,

researchers assume that fibres are arranged in various geometrical arrays (triangular,

rectangular, hexagonal etc.) or they are distributed randomly in the cross-section In both

cases the composite can be assumed as isotropic in the cross-sectional plane However,

anisotropic materials are also very common What is more, one may intentionally create

composite because of desired resultant properties of such materials The influence of

topological configuration of fibres in unidirectional composite is shown at Figs 2A-2C The

plot (Fig 2C) shows the relation between the effective thermal conductivity and the angle β

by which fibres are rotated from horizontal to vertical alignment

The minimal value of effective thermal conductivity is shown at Fig 2B, maximal value at

3 Numerical procedures

Numerical calculations were performed by hybrid method which consisted of two

procedures: finite element method used for solving differential equation and genetic

algorithm for optimization Both procedures were implemented in COMSOL Script

3.1 Finite element method (FEM)

A case in which heat transfer can be considered to be adequately described by a

two-dimensional formulation is shown in Fig 3 Two two-dimensional steady heat transfer in

considered domain is governed by following heat transfer equation:

0, (12)

in the domain Ω

1 All figures in this paper presenting the elementary composite cell use the same sizes and the same

temperature scale as figures Fig 2A and Fig 2B, so the scales are omitted on the next figures Isolines are

presented in reversed grayscale

(a) (b)

(c)

Fig 3 Geometry of domain with boundary conditions

In the considered problem one can take under consideration three types of heat transfer

heat transfer coefficient, – thermal conductivity coefficient, nx and ny – components of

normal vector to boundary

In developing a finite element approach to dimensional conduction we assume a

two-dimensional element having M nodes such that the temperature distribution in the element

is described by

row matrix of interpolation functions, and {T} is the column matrix (vector) of nodal

temperatures

Applying Galerkin’s finite element method (Zienkiewicz&Taylor, 2000), the residual

equations corresponding to steady heat transfer equation are

Using Green’s theorem in the plane we obtain

(18)and by transforming left-hand side we obtain:

in the Galerkin residual equation we obtain

(24)

The equation (24) we can rewrite for the whole considered domain which gives us the

following matrix equation

(25)

where K is the conductance matrix, a is the solution for nodes of elements, and f is the

forcing functions described in column vector

The conductance matrix

(26) and the forcing functions

(27) are described by following integrals

, , (31)

Equations 25-32 represent the general formulation of a finite element for two-dimensional

heat conduction problem In particular these equations are valid for an arbitrary element

having M nodes and, therefore, any order of interpolation functions Moreover, this

formulation is valid for each composite constituent

3.2 Genetic algorithm (GA)

Genetic algorithm is one of the most popular optimization techniques (Koza, 1992) It is

based on an analogy to biological mechanism of evolution and for that reason the

terminology is a mixture of terms used in optimization and biology Optimization in a

simple case would be a process of finding maximum (or minimum) value of an objective

function:

In GA each potential solution is called an individual whereas the space of all the feasible

values of solutions is a search space Each individual is represented in its encoded form,

called a chromosome The objective function which is the measure of quality of each

chromosome in a population is called a fitness function The optimization problem can be

expressed in the following form:

where: x denotes the best solution, f is an objective function, x represents any feasible

solution and D is a search space Chromosomes ranked with higher fitness value are more

likely to survive and create offspring and the one with the highest value is taken as the best

solution to the problem when the algorithm finishes its last step The concept of GA is

presented at fig 4

Algorithm starts with initial population that is chosen randomly or prescribed by a user As

GA is an iterative procedure, subsequent steps are repeated until termination condition is

satisfied The iterative process in which new generations of chromosomes are created

involves such procedures as selection, mutation and cross-over Selection is the procedure

used in order to choose the best chromosomes from each population to create the new

generation Mutation and cross-over are used to modify the chromosomes, and so to find

new solutions GA is usually used in complex problems i.e high dimensional,

multi-objective with multi connected search space etc Hence, it is common practice that users

search for one or several alternative suboptimal solutions that satisfy their requirements,

rather than exact solution to the problem In this paper GA optimizes geometrical

arrangement of fibres in a composite materials as it influences effective thermal conductance

of this composite It has been developed many improvements to the original concept of GA

introduced by Holland (Holland, 1975) such as floating point chromosomes, multiple point

crossover and mutation, etc However, binary encoding is still the most common method of

encoding chromosomes and thus this method is used in our calculations

3.2.1 Encoding

We consider an elementary cell of a composite that is 2-D domain and there are N fibres

inside the cell, the position of each fibre is defined by its coordinates, which means we need

Fig 4 Genetic algorithm scheme

to optimize 2N variables Furthermore, it is assumed that each coordinate is determined

and upper limit of the range respectively It means that each domain needs to be divided

encode variables:

Consequently, we can calculate the number of bits required to encode a chromosome:

In our calculation we assume three significant digits precision which means we need 2 bits

to encode each variable

3.2.2 Fitness and selection

Selection is a procedure in which parents for the new generation are chosen using the fitness

function There are many procedures possible to select chromosomes which will create

another population The most common are: roulette wheel selection, tournament selection,

rank selection, elitists selection

In our case, modified fitness proportionate selection also called roulette wheel selection is

of being selected is calculated for every individual chromosome Consequently, the

candidate solution whose fitness is low will be less likely selected as a parent whereas it is

more probable for candidates with higher fitness to become a parent The probability of

selection is determined as follows:

where S is the number of chromosomes in population

Modification of the roulette wheel selection that we introduced is caused by the fact that we needed to perform constrained optimization The constrains are the result of the fact that fibres cannot overlap with each other There are some possible options to handle this problem, one of which would to use penalty function During calculations, however, it turned out that this approach is less effective than the other one based on elitist selection

We decided that in case of chromosome representing arrangement of overlapping fibres such chromosome should be replaced with the best one

3.2.3 Genetic operators

Cross-over operation requires two chromosomes (parents) which are cut in one, randomly chosen point (locus) and since this point the binary code is swapped between the chromosomes creating two, new chromosomes, as it is shown at Fig 5

Mutation procedure in case of binary representation of solution is an operation of bit inversion at randomly chosen position Fig6 The following purpose of this procedure is

to introduce some diversity into population and so to avoid premature convergence to local maximum

Fig 5 Crossover procedure scheme

Fig 6 Mutation procedure scheme

4 Numerical results

All optimization problems considered in this chapter are governed by Eq 6 for each constituent of the composite with appropriate boundary conditions (7-11) In our calculations we assumed the same sizes of the unit cell i.e 1x1cm ( Fig1.) Temperatures on

calculation were made using second order triangular Lagrange elements The stationary problem of heat transfer was solved using direct UMFPACK linear system solver The mesh structure depends on the number and positions of fibres and so the number of mesh elements was not larger than 5000

We performed three types of optimization in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives desired value

of effective thermal conductivity In the latter case we defined the objective function as the minimization of the deviation from the expected value The results of optimization are presented at Figs 7-9

4.1 Optimization of three and four fibres arrangement

In the beginning we assumed the same sizes of the fibres, as well as the same value of thermal conductivity for each fibre Numerical values of parameters used in calculations, and the resultant effective thermal conductivity was shown in Table 1 The ‘Opt.’ column

results agree with results presented in section 2.3 Figures 7A and 7E present the arrangement obtained during minimization All fibres are aligned horizontally perpendicularly to heat flux direction, next to each other In case of maximization (Figs 7B, 7F) fibres are aligned vertically – along with heat flux direction

However, there are many possible ways of arrangement of intermediate values of effective thermal conductivity – fibres do not have to be aligned anymore as it was assumed at Fig 2C We also presented one of possible arrangements that result in a composite with effective thermal conductivity equal to the one expected for each number of fibres: (Figs 7C, 7D) If one would like to achieve certain value of effective thermal conductivity with respect to some geometrical assumptions (for instance minimum/maximum distance between fibres)

it is also possible to perform such optimization, however penalty function should be implemented or objective function modified to include such conditions

4.2 Optimization of five and six fibres arrangement

Calculation performed for five and six fibres were similar to those presented above for three and four fibres However, the more fibres the more complex problem As it was mentioned

in section 3.2.1 each fibre is described by two variables changing within the range [0,1] with

direct impact on calculation time and so it takes far more time to find optimal solution The terminating condition of GA was set to 2000 iterations for three and four fibres It resulted in almost perfect arrangement in case of three fibres whereas the arrangement for four fibres was not equally well While increasing the number of fibres to five and six fibres,

we also increased the number of iteration to 10000

Another important aspect of the considered problem was that in case of five and six fibres of assumed radii (Table 2) it was not possible to align them in one row so the relation presented in section 2.3 could not be applied anymore

The minimization results for five and six fibres were presented at Figs 8A and 8E, the maximization results at Figs 8B and 8F and the arrangement for expected value of effective

thermal conductivity at Figs 8C, 8D One can notice that the arrangement of fibres is also close to horizontal in case of minimization and close to vertical in case of maximization, although fibres are not localised next to each other and initialization of the second row in case of six fibres can be observed In general, however, we may not assume that fibres are always aligned in rows in case of minimum and maximum values of effective thermal conductivity The situation changes when the thermal conductivity of fibres is not the same

in each fibre The result for such situation was presented in the next section

4.3 Optimization of four and five fibres arrangement with different radii and thermal conductivity of fibres

Apart from the simplest case in which the composite consisted of identical fibres we also analysed the case in which fibres differ from each other We used two sizes of fibres with different values of thermal conductivities All parameters used in calculations were presented

different radii and thermal conductivities

We performed the optimization of the arrangement of four and five fibres in a composite cell The minimization results were presented at Figs 9A, 9E while maximization at Figs 9B, 9F The arrangements obtained for the assumed values of effective thermal conductivity for four and five fibres were presented at Figs 9C,9D respectively It is remarkable, that in these

cases the optimal arrangement of fibres is no longer that predictable Fibres are not aligned

in a row, although there was enough space However, fibres still tend to be close to each other but spatial configuration is changed

5 Conclusion

This study has examined the effect of multi fibres filler in composite on thermal conductivity Three types of optimization were performed in terms of effective thermal conductivity: minimization, maximization and determination of arrangement which gives expected value of effective thermal conductivity Hybrid method combining optimization with genetic algorithm and differential equation solver by finite element method were used

to find optimal arrangement of fibres position in composite matrix was used in this work Proposed algorithm was implemented in Comsol Multiphysics environment

It was proved that the geometrical structure of the composite (matrix and filler arrangement) may have a great impact on the resultant effective conductivity of the composite In many research works it is assumed that fibres are arranged in various geometrical arrays or they are distributed randomly in the cross-section

Through this study, some areas were found that need to be investigated further Composite constituents can be anisotropic, and with temperature dependent thermal conductivity of constituents (e.g resin, fibre, void)

6 References

Al-Sulaiman FA, Mokheimer EM, Al-Nassar YN (2006) Prediction of the thermal

conductivity of the constituents of fiber reinforced composite laminates, Heat Mass

Transfer, 42, 5, 370–377

Al-Sulaiman FA, Al-Nassar YN and Mokheimer EM (2006) Composite Laminates: Voids

Effect Prediction of the Thermal Conductivity of the Constituents of

Fiber-Reinforced, Journal of Composite Materials, 40; pp.797-814

Boguszewski T., Ciupiński Ł., Kurzydłowski K.J (2008) Numerical Calculation of The

Thermal Conductivity Coefficient in Diamond-Copper Composite,

Kompozyty/Composites, Vol 8, Nr 3, pp.232-235

Brucker Kyle A., Majdalani Joseph (2005) Effective thermal conductivity of common

geometric shapes, International Journal of Heat and Mass Transfer, 48, pp.4779–4796 Comsol Multiphysics User’s Guide (2007) Modeling Guide and Model Library, Documentation

Set, Comsol AB

Duc N.D., Boi L.V., Dac N.T (2008) Determining thermal expansion coefficients of

three-phase fiber composite material reinforced by spherical particles, VNU Journal of

Science, Mathematics – Physics, 24, pp.57-65

Freger G.E., Kestelman V.N., Freger D.G (2004) Spirally Anisotropic Composites,

Springer-Verlag Berlin Heidelberg

Holland J H (1975) Adaptation in natural and artificial systems, The University Michigan

Press Ann Arbor

Kidalov Sergey V., Shakhov Fedor M (2009) Thermal Conductivity of Diamond

Composites, Materials, 2, pp.2467-2495